Problem 1351

(a) Does the game necessarily end?

1. Let $N$ denote the total number of cards. We need to show that the game ends for any $N$.
2. For $N = 1$, the game ends trivially because there are no blocks of 50 cards to flip.
3. Assume the game ends for $1, 2, 3, \ldots, N$ cards. We need to show that the game also ends for $N+1$ cards.
4. Consider the cards numbered $1, 2, 3, \ldots, N, N+1$.

Case 1: If card 1 is black-sided, then we can choose the other $N$ cards. By the inductive hypothesis, the game will end for these $N$ cards.

Case 2: If card 1 is gold-sided, we have two sub-cases:

Sub-Case 1: We never choose card 1. Then, by the inductive hypothesis, the game ends for the remaining $N$ cards.

Sub-Case 2: If we choose card 1, it becomes black-sided, and we are left with $N$ cards. By the inductive hypothesis, the game will end for these $N$ cards.

5. By induction, the game necessarily ends for any number of cards $N$.

$\blacksquare$

(b) Does there exist a winning strategy for the starting player?

1. Consider the cards at positions $50, 100, 150, \ldots, 2000$. Initially, all of these cards are gold-sided.
2. After the first player's move, no matter how they choose 50 consecutive cards, there must be at least one card from the set $\{50, 100, 150, \ldots, 2000\}$ that turns black.
3. When the second player moves, there will be an odd number of cards from the set $\{50, 100, 150, \ldots, 2000\}$ that are black-sided.
4. The second player can always find a move because there will always be a block of 50 cards starting with a gold-sided card.
5. As the game progresses, the first player will eventually be unable to make a legal move, meaning the second player will win.
6. Therefore, the first player does not have a winning strategy.

$\blacksquare$